Learn Swedish: Den nya skärmen har en ännu skarpare bild. - The new screen has an even sharper picture.

Ännu means "even" when used with a comparative adjective. So, ännu skarpare means "even sharper". It emphasizes that the previous screen was already sharp, but this new one takes it to the next level.

Could I use mycket instead of ännu here?

Yes, but it changes the meaning slightly. En mycket skarpare bild translates to "a much sharper picture". Using ännu specifically means "even sharper".

Would the ending of skarpare change if the noun was an ett word or plural?

No. A great thing about Swedish comparative adjectives (like skarpare) is that they are invariable. They never change their endings to match the noun's gender or number. You would say ett ännu skarpare foto (an even sharper photo) exactly the same way.

Why is the subject Den nya skärmen and not just Den ny skärm?

This is a classic example of Swedish "double definiteness", which you learn earlier on but is easy to forget. Because we are talking about a specific screen with an adjective attached, we need three definite markers: the front article Den, the definite adjective form nya (ending in -a), and the definite noun suffix on skärmen.